The deterministic FitzHugh-Nagumo system is an important mathematical model to describe the signal transmission across axons in neurobiology, which is well studied in the literature, see, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
In the stochastic case,
Wang [7]} proved the existence and uniqueness of random attractors in the initial space \(L^2(\mathbb {R}^N)\times L^2(\mathbb {R}^N)\)  for the autonomous FitzHugh-Nagumo system. For the general non-autonomous forcings \(g\)  and \(h\) , Adili and Wang [8]}, [9]} obtained the pullback attractors in the initial space \(L^2(\mathbb {R}^N)\times L^2(\mathbb {R}^N)\) , and Bao [10]} strengthened this result and proved the regularity of pullback attractors in the non-initial space \(H^1(\mathbb {R}^N)\times L^2(\mathbb {R}^N)\) . Recently, [11]}, [12]}, [13]} studied the existences of random attractors for stochastic lattice FitzHugh-Nagumo system
